Higher category theory
higher category theory Basic concepts
Extra properties and structure
The notion of
3-groupoid is the next higher generalization in higher category theory of groupoid and 2-groupoid. Definition
3-groupoid is an ∞-groupoid such that all parallel pairs of k-morphism are equivalent for : a 3- k ≥ 4 k \geq 4 truncated ∞-groupoid.
Thus, up to
equivalence, there is no point in mentioning anything beyond -morphisms, except whether two given parallel 3 3 -morphisms are equivalent. This definition may give a concept more general than your preferred definition of 3 3 -groupoid, but it will be equivalent; basically, you may have to rephrase equivalence of 3 3 -morphisms as 3 3 equality.
A general 3-groupoid is
geometrically modeled by a 4- coskeletal Kan complex. Equivalently – via the homotopy hypothesis-theorem – by a homotopy 3-type.
A small model of this is a 3-
hypergroupoid, where all horn-filelrs in dimension are ≥ 4 \geq 4 unique .
A 3-groupoid is
algebraically modeled by a tricategory in which all morphisms are invertible, and by a 3- truncated algebraic Kan complex.
semistrict algebraic model for 3-groupoids is provided by the notion of Gray-groupoid. These in turn are encoded by 2-crossed modules.
An entirely strict algebraic model for 3-groupoids (which no longer models all
homotopy 3-types) is a 3- truncated strict omega-groupoid.
h-level 2 | 0-truncated |
discrete space | 0-groupoid/ set | sheaf | h-set h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/ groupoid | (2,1)-sheaf/ stack | h-groupoid h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | | h-2-groupoid h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | | h-3-groupoid h-level | n + 2 n+2 -truncated | n n homotopy n-type | n-groupoid | | h- -groupoid | h-level n n | untruncated | ∞ \infty homotopy type | ∞-groupoid | (∞,1)-sheaf/ ∞-stack | h- -groupoid ∞ \infty References
Simona Paoli, Semistrict models of connected 3-types and Tamsamani’s weak 3-groupoids,
Journal of Pure and Applied Algebra 211 (2007), 801-820. ( arXiv)
Carlos Simpson, Homotopy types of strict 3-groupoids ( arXiv)
Revised on September 10, 2012 20:26:30